Dean's Office Official Model Question Paper

Type I and Type II Errors

• Type I Error ($\alpha$): Rejecting the null hypothesis ($H_0$) when it is actually true (false positive). The probability of committing a Type I error is the significance level $\alpha$.
• Type II Error ($\beta$): Failing to reject the null hypothesis ($H_0$) when it is actually false (false negative). The probability of correctly rejecting a false null hypothesis is statistical power ($1 - \beta$).

Laspeyre’s vs. Paasche’s Price Index

• Laspeyre’s Index ($P_{01}^L$): Uses base-year quantities ($q_0$) as weights: $P_{01}^L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$. It tends to overestimate price increases (upward bias) because it ignores consumer substitution toward cheaper goods.
• Paasche’s Index ($P_{01}^P$): Uses current-year quantities ($q_1$) as weights: $P_{01}^P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$. It tends to underestimate price increases (downward bias) by over-weighting goods whose relative prices have fallen.

Gauss-Markov Assumptions of CLRM

1. Linearity in Parameters: The model is linear in coefficients: $Y = \beta_0 + \beta_1 X + \epsilon$.
2. Zero Conditional Mean: $E(\epsilon \mid X) = 0$ (error disturbances have expected mean zero).
3. Homoskedasticity: $\text{Var}(\epsilon_i \mid X) = \sigma^2$ (constant error variance across all observations).
4. No Autocorrelation: $\text{Cov}(\epsilon_i, \epsilon_j) = 0$ for $i \ne j$ (uncorrelated error terms).
5. Exogeneity: $\text{Cov}(X_i, \epsilon_i) = 0$ (independent variables are uncorrelated with error terms).

When these hold, OLS estimators are BLUE (Best Linear Unbiased Estimators).

$R^2$ and Adjusted $\bar{R}^2$

• $R^2$ (Coefficient of Determination): Measures the proportion of total variation in the dependent variable explained by the regression model: $R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$.
• Need for $\bar{R}^2$: In multiple regression, $R^2$ mechanically increases whenever any additional independent variable is added, even if irrelevant. Adjusted $\bar{R}^2$ penalizes the addition of non-significant variables by adjusting for degrees of freedom: $\bar{R}^2 = 1 - \left(1 - R^2\right) \frac{n - 1}{n - k - 1}$.

Five-Number Summary of Box and Whisker Plot

A Box and Whisker plot visualizes distribution, central tendency, and dispersion using:

1. Minimum Value: Lowest data point within the lower fence ($Q_1 - 1.5 \times IQR$).
2. First Quartile ($Q_1$): 25th percentile (bottom of the box).
3. Median ($Q_2$): 50th percentile (central line within the box).
4. Third Quartile ($Q_3$): 75th percentile (top of the box).
5. Maximum Value: Highest data point within the upper fence ($Q_3 + 1.5 \times IQR$). Data points outside the fences are identified as outliers.

Numerical Problem: Independent Two-Sample t-Test

1. Hypotheses Formulation

• Null Hypothesis ($H_0$): $\mu_1 - \mu_2 = 0$ (There is no significant difference in mean processing turnaround times between Pokhara and Biratnagar hubs).
• Alternative Hypothesis ($H_1$): $\mu_1 - \mu_2 \ne 0$ (There is a statistically significant difference; two-tailed test).
• Significance level: $\alpha = 0.05$.
• Degrees of freedom: $df = n_1 + n_2 - 2 = 12 + 10 - 2 = 20$.

2. Calculation of Pooled Variance ($s_p^2$)

$$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$$

$$s_1^2 = (2.4)^2 = 5.76$$

$$s_2^2 = (2.8)^2 = 7.84$$

$$s_p^2 = \frac{(12 - 1)(5.76) + (10 - 1)(7.84)}{20} = \frac{11(5.76) + 9(7.84)}{20} = \frac{63.36 + 70.56}{20} = \frac{133.92}{20} = 6.696$$

$$s_p = \sqrt{6.696} \approx 2.5877\text{ hours}$$

3. Standard Error of the Difference ($SE$)

$$SE = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} = 2.5877 \times \sqrt{\frac{1}{12} + \frac{1}{10}} = 2.5877 \times \sqrt{0.08333 + 0.1000}$$

$$SE = 2.5877 \times \sqrt{0.18333} = 2.5877 \times 0.42817 \approx 1.108\text{ hours}$$

4. Calculated t-Statistic ($t_{cal}$)

$$t_{cal} = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{14.5 - 17.2}{1.108} = \frac{-2.7}{1.108} \approx \mathbf{-2.437}$$

$$|t_{cal}| = 2.437$$

5. Decision & Managerial Conclusion

• Critical Value: For $df = 20$ at $\alpha = 0.05$ (two-tailed), $t_{crit} = 2.086$.
• Decision Rule: Reject $H_0$ if $|t_{cal}| > t_{crit}$.
• Since $|t_{cal}| = 2.437 > 2.086$, we reject $H_0$ at the 5% significance level ($p < 0.05$).
• Conclusion: There is a statistically significant difference in SME loan turnaround times between Pokhara and Biratnagar hubs. Pokhara operates significantly faster (14.5 hours vs. 17.2 hours). Bank management should investigate procedural bottlenecks and underwriting delays in Biratnagar to standardize processing velocity.

Numerical Problem: Chi-Square Test of Independence

1. Hypotheses Formulation

• Null Hypothesis ($H_0$): Customer channel preference (Mobile App vs. Physical Branch) is independent of age group.
• Alternative Hypothesis ($H_1$): Customer channel preference is dependent on age group.
• Significance level: $\alpha = 0.05$.
• Degrees of freedom: $df = (r - 1)(c - 1) = (3 - 1)(2 - 1) = 2 \times 1 = 2$.

2. Calculation of Expected Frequencies ($E_{ij} = \frac{R_i \times C_j}{N}$)

• Row 1 (Young, $R_1 = 120$):
  • $E_{11} (\text{Mobile}) = \frac{120 \times 180}{300} = 72.0$
  • $E_{12} (\text{Branch}) = \frac{120 \times 120}{300} = 48.0$
• Row 2 (Middle-Aged, $R_2 = 110$):
  • $E_{21} (\text{Mobile}) = \frac{110 \times 180}{300} = 66.0$
  • $E_{22} (\text{Branch}) = \frac{110 \times 120}{300} = 44.0$
• Row 3 (Senior, $R_3 = 70$):
  • $E_{31} (\text{Mobile}) = \frac{70 \times 180}{300} = 42.0$
  • $E_{32} (\text{Branch}) = \frac{70 \times 120}{300} = 28.0$

3. Chi-Square Test Statistic Computation ($\chi^2 = \sum \frac{(O - E)^2}{E}$)

$$\chi^2_{cal} = \mathbf{28.490}$$

4. Decision & Strategic Implications

• Critical Value: For $df = 2$ at $\alpha = 0.05$, $\chi^2_{crit} = 5.991$.
• Decision: Since $\chi^2_{cal} = 28.490 > 5.991$, we reject $H_0$ at the 5% significance level ($p < 0.001$).
• Conclusion: Banking channel preference is highly dependent on age. Younger clients disproportionately adopt digital mobile banking ($75\%$ vs. expected $60\%$), whereas senior citizens heavily rely on physical branch interactions ($64.3\%$ vs. expected $40\%$). The bank should maintain hybrid branch services while designing targeted digital onboarding and accessibility features for seniors.

Econometric Diagnostics: Multicollinearity and Heteroskedasticity

In multiple linear regression $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \dots + \beta_k X_k + \epsilon$, departures from CLRM assumptions undermine statistical inference:

+-----------------------+------------------------+-------------------------------+
| Diagnostic Dimension  | Multicollinearity      | Heteroskedasticity            |
+-----------------------+------------------------+-------------------------------+
| Violated Assumption   | Regressors independent | Error variance constant       |
| Nature of Violation   | High linear correlation| Non-constant variance Var(e|X)|
| OLS Estimator Quality | Unbiased but large SE  | Unbiased but inefficient      |
| Hypothesis Tests      | Artificially low t-stat| Biased SE; invalid t/F tests  |
+-----------------------+------------------------+-------------------------------+

1. Multicollinearity

• Causes: Occurs when two or more explanatory variables share strong linear relationships (e.g., including both household disposable income and wealth in a loan demand model).
• Consequences:
  • OLS coefficients remain unbiased and consistent, but standard errors become inflated.
  • Individual t-statistics are insignificant despite a very high overall $R^2$ and highly significant F-statistic.
  • Estimates are hypersensitive to small data sample alterations.
• Detection Methods:
  • Correlation matrix inspection ($r > 0.80$).
  • Variance Inflation Factor (VIF): $VIF_j = \frac{1}{1 - R_j^2}$. A rule of thumb is $VIF > 5$ to $10$ signifies problematic collinearity.
• Remedies: Drop redundant collinear variables; transform variables using first differences or ratios; employ Principal Component Analysis (PCA) or Ridge regression.

2. Heteroskedasticity

• Causes: Occurs when error variance varies across observations: $\text{Var}(\epsilon_i) = \sigma_i^2$ (common in cross-sectional firm-level banking data where large banks exhibit greater variance than small microfinance institutions).
• Consequences:
  • OLS estimates remain unbiased, but they are no longer Best (efficient).
  • Standard errors are biased, rendering usual confidence intervals and t/F tests invalid.
• Detection Methods:
  • Residual scatter plots ($e_i^2$ against $\hat{Y}_i$).
  • Formal statistical tests: Breusch-Pagan-Godfrey test and White’s General Heteroskedasticity test.
• Remedies:
  • Logarithmic transformations of non-linear financial variables (e.g., $\ln(\text{Assets})$).
  • Estimation using White’s Heteroskedasticity-Consistent Robust Standard Errors.
  • Weighted Least Squares (WLS) where weights are inversely proportional to estimated error variances.

(a) Fisher’s Ideal Index Number & Consistency Tests

Fisher’s Ideal Index is the geometric mean of Laspeyre’s ($L$) and Paasche’s ($P$) price index numbers:

$$P_{01}^F = \sqrt{P_{01}^L \times P_{01}^P} = \sqrt{\left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right)} \times 100$$

1. Time Reversal Test ($P_{01} \times P_{10} = 1$)

Interchanging base year (0) and current year (1):

$$P_{10}^F = \sqrt{\left(\frac{\sum p_0 q_1}{\sum p_1 q_1}\right) \times \left(\frac{\sum p_0 q_0}{\sum p_1 q_0}\right)}$$

Multiplying both price indices (excluding factor 100):

$$P_{01}^F \times P_{10}^F = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum p_0 q_1}{\sum p_1 q_1} \times \frac{\sum p_0 q_0}{\sum p_1 q_0}} = \sqrt{1} = \mathbf{1}$$

Hence, Fisher’s Index satisfies the Time Reversal Test.

2. Factor Reversal Test ($P_{01} \times Q_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0}$)

Interchanging price ($p$) and quantity ($q$) to obtain the quantity index:

$$Q_{01}^F = \sqrt{\left(\frac{\sum q_1 p_0}{\sum q_0 p_0}\right) \times \left(\frac{\sum q_1 p_1}{\sum q_0 p_1}\right)} = \sqrt{\left(\frac{\sum p_0 q_1}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_1 q_1}{\sum p_1 q_0}\right)}$$

Multiplying price and quantity indices:

$$P_{01}^F \times Q_{01}^F = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum p_0 q_1}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_1 q_0}} = \sqrt{\frac{(\sum p_1 q_1)^2}{(\sum p_0 q_0)^2}} = \mathbf{\frac{\sum p_1 q_1}{\sum p_0 q_0}}$$

Hence, Fisher’s Index satisfies the Factor Reversal Test (true value ratio).

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(b) Components of Financial Time Series

A time series decomposes into four systematic elements under the multiplicative model $Y = T \times S \times C \times I$:

1. Secular Trend ($T$): Long-term smooth progression over decades (e.g., secular growth in national banking assets).
2. Seasonal Variation ($S$): Short-term regular quarterly or monthly recurring fluctuations within a year (e.g., surge in currency in circulation before Dashain festival).
3. Cyclical Movement ($C$): Medium-term wave-like oscillations spanning 3 to 10 years tied to macroeconomic boom and recession business cycles.
4. Irregular / Random Disturbance ($I$): Unpredictable shocks (e.g., natural disasters, geopolitical supply shocks).

Seasonal Index Calculation: The Ratio-to-Moving-Average method computes a 4-quarter centered moving average to eliminate trend and cycle ($T \times C$), isolates seasonal-irregular ratios ($S \times I = \frac{Y}{T \times C}$), and averages quarterly values to yield clean Seasonal Indices normalized to $400\%$.

Case Analysis: Determinants of Commercial Bank Credit Growth

(a) Interpretation of Regression Coefficients (5 Marks)

1. Intercept ($\hat{\beta}_0 = 15.40$):
   • When total deposits are zero ($X_1 = 0$) and base interest rate is zero ($X_2 = 0$), the baseline quarterly bank loan disbursement is estimated at NPR 15.40 Billion. Statistically, it represents autonomous credit disbursements funded via equity capital or interbank borrowings.
2. Slope of Deposits ($\hat{\beta}_1 = +0.72$):
   • Holding the base rate constant, for every NPR 1.0 Billion increase in total deposits, loan disbursements increase on average by NPR 0.72 Billion (720 Million).
   • Economic Consistency: Positively signed, aligning with commercial banking theory and NRB’s Credit-Deposit (CD) prudential framework where deposit mobilization directly expands lending capacity.
3. Slope of Base Interest Rate ($\hat{\beta}_2 = -1.85$):
   • Holding deposit volume constant, for every 1.0 percentage point (100 bps) increase in the base lending rate, loan disbursements decrease on average by NPR 1.85 Billion.
   • Economic Consistency: Negatively signed, consistent with the classical law of demand for credit; higher borrowing costs deter corporate and consumer capital expenditures.

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(b) Goodness-of-Fit Metrics & Standard Error of Estimate (5 Marks)

1. Coefficient of Determination ($R^2$):

$$R^2 = \frac{SSR}{SST} = \frac{3,600}{4,500} = \mathbf{0.8000\ (80.0\%)}$$

• Interpretation: 80.0% of the total quarterly variation in commercial bank loan disbursements is explained by deposit volumes and base lending rates.

2. Adjusted Coefficient of Determination ($\bar{R}^2$):

$$\bar{R}^2 = 1 - \left(1 - R^2\right) \left(\frac{n - 1}{n - k - 1}\right)$$

Where sample size $n = 30$ and number of explanatory variables $k = 2$ ($df = n - k - 1 = 30 - 2 - 1 = 27$):

$$\bar{R}^2 = 1 - (1 - 0.8000) \times \left(\frac{29}{27}\right) = 1 - (0.2000 \times 1.07407) = 1 - 0.21481 = \mathbf{0.7852\ (78.52\%)}$$

• Interpretation: After adjusting for degrees of freedom, the model reliably accounts for 78.52% of variance, confirming high predictive power without overfitting.

3. Standard Error of the Estimate ($s_e$):

$$s_e = \sqrt{\frac{SSE}{n - k - 1}} = \sqrt{\frac{900}{27}} = \sqrt{33.333} \approx \mathbf{5.7735\text{ NPR Billion}}$$

• Interpretation: The standard deviation of observed loan disbursements around the fitted regression hyperplane is approximately NPR 5.77 Billion.

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(c) Individual Hypothesis Testing ($t$-Tests) at $\alpha = 0.05$ (5 Marks)

• Degrees of freedom: $df = n - k - 1 = 30 - 2 - 1 = 27$.
• Two-tailed critical value: $t_{0.025, 27} = 2.052$.

1. Test for Total Deposits ($\beta_1$):

• Hypotheses: $H_0: \beta_1 = 0$ vs. $H_1: \beta_1 \ne 0$$t_{cal, 1} = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} = \frac{0.72}{0.09} = \mathbf{+8.000}$$
• Decision: Since $|t_{cal, 1}| = 8.000 > 2.052$, we reject $H_0$ ($p < 0.0001$). Total customer deposits exert a statistically significant positive effect on bank loan growth.

2. Test for Base Lending Rate ($\beta_2$):

• Hypotheses: $H_0: \beta_2 = 0$ vs. $H_1: \beta_2 \ne 0$$t_{cal, 2} = \frac{\hat{\beta}_2 - 0}{SE(\hat{\beta}_2)} = \frac{-1.85}{0.62} = \mathbf{-2.984}$$
  $$|t_{cal, 2}| = 2.984$$
• Decision: Since $|t_{cal, 2}| = 2.984 > 2.052$, we reject $H_0$ ($p < 0.01$). The base interest rate exerts a statistically significant negative effect on loan disbursements.

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(d) Overall Model Significance ($F$-Test) & Loan Forecast (5 Marks)

1. Overall $F$-Test for Goodness of Fit:

• Hypotheses: $H_0: \beta_1 = \beta_2 = 0$ (Model has no explanatory power) vs. $H_1: \text{At least one } \beta_j \ne 0$.
• Mean Square Regression ($MSR$): $MSR = \frac{SSR}{k} = \frac{3,600}{2} = 1,800$.
• Mean Square Error ($MSE$): $MSE = \frac{SSE}{n - k - 1} = \frac{900}{27} = 33.333$.
  $$F_{cal} = \frac{MSR}{MSE} = \frac{1,800}{33.333} = \mathbf{54.00}$$
• Critical Value: $F_{0.05, (2, 27)} = 3.35$.
• Decision: Because $F_{cal} = 54.00 > 3.35$, we reject $H_0$ ($p < 0.0001$). The multiple regression model as a whole is exceptionally robust and statistically significant.

2. Point Prediction for $X_1 = 150\text{ Billion}$, $X_2 = 8.5\%$:

$$\hat{Y} = 15.40 + 0.72(150) - 1.85(8.5)$$

$$\hat{Y} = 15.40 + 108.00 - 15.725 = \mathbf{107.675\text{ NPR Billion}}$$

• Forecasting Conclusion: For a bank holding NPR 150 Billion in deposits under an 8.5% base rate environment, the expected loan portfolio volume is NPR 107.68 Billion.